A simple disturbance observer for stator flux linkage estimation of nonlinear synchronous machines

Synchronous machines (SMs) like reluctance synchronous machines (RSMs) or interior permanent magnet synchronous machines (IPMSMs) are characterized by nonlinear stator flux linkages which are crucial for optimal control and operation management. This paper presents a novel disturbance observer that allows to estimate the nonlinear flux linkages online for any SM (with constant or no excitation). Moreover, a simple identification sequence with ramp-like voltage signals is proposed to extract key information of the nonlinear stator flux linkage maps within a short period of time and without the need of any controller, torque sensor or prime mover. Finally, the proposed observer with identification sequence is implemented and validated for a real and highly nonlinear IPMSM by realistic and detailed simulation results.

A simple disturbance observer for stator flux linkage estimation of nonlinear synchronous machines Niklas Monzen * , Bernd Pfeifer * and Christoph M. Hackl * Laboratory for Mechatronic and Renewable Energy Systems (LMRES) HM Munich University of Applied Sciences Munich, Germany {niklas.monzen, bernd.pfeifer,christoph.hackl}@hm.eduAbstract-Synchronous machines (SMs) like reluctance synchronous machines (RSMs) or interior permanent magnet synchronous machines (IPMSMs) are characterized by nonlinear stator flux linkages which are crucial for optimal control and operation management.This paper presents a novel disturbance observer that allows to estimate the nonlinear flux linkages online for any SM (with constant or no excitation).Moreover, a simple identification sequence with ramp-like voltage signals is proposed to extract key information of the nonlinear stator flux linkage maps within a short period of time and without the need of any controller, torque sensor or prime mover.Finally, the proposed observer with identification sequence is implemented and validated for a real and highly nonlinear IPMSM by realistic and detailed simulation results.
Index Terms-Cross-coupling, flux observer, nonlinear machine model, parameter identification, permanent magnet synchronous machine, reluctance synchronous machine, saturation.

I. MOTIVATION
Many control approaches of electrical drive systems are based on linear [1] or nonlinear [2,3] electrical machine models which rely on machine parameters (e.g.resistance) and apparent or differentiable inductances and/or flux linkage maps.Moreover, also for detailed dynamic closed-loop simulations of the considered electrical drive system, accurate models capturing the nonlinear dynamics and losses of the machine are crucial.The electromagnetic behavior of RSMs and (I)PMSMs are characterized by saturation and cross-coupling effects.In this regards, in particular, flux linkage maps play the key role for accurate machine modeling [4], advanced current control [5], optimal feedforward torque control [6,7] or rotor position estimation [8,9].This paper focus on the identification of the nonlinear stator flux linkages of any SMs (with constant or no excitation), such as reluctance synchronous machines (RSMs) and permanent magnet synchronous machines with surface mounted magnets (SPMSMs) or with interior magnets (IPMSMs).If the design of a machine is known, the electromagnetic machine behavior can be extracted from finite element analysis (FEA) [10].However, usually the FEA data is not available at all or, if available, manufacturing and material tolerances are difficult to be considered.Consequently, experimental identification technics are needed in any case for machine identification.
* All authors contributed equally to the paper.
A comprehensive overview on machine identification methods is given in [11].Some papers are discussed in more detail in the following.In [12][13][14], a steady-state test is performed in order to obtain the nonlinear flux linkage maps.All of these approaches need a stator current controller to ensure steady-state currents.However, in general, controller design and its tuning are tedious as the machine dynamics are actually not known a priori.Furthermore, in [13], a test bench with additional prime mover is needed, whereas, in [14], an accurate current controller to compensate for variable speeds is required.In general, such approaches are time-consuming because the respective quantities such as currents, speed and flux linkages are extracted at steady state and filtered (averaged) over several milliseconds to mitigate noise and measurement inaccuracies.In [15,16], a fast identification procedure by integrating the time derivative of the flux linkages is presented.However, it is only suitable for RSMs.For PMSM, the initial flux linkage value is usually not known as the permanent magnet flux linkage acts as unknown offset.By the high frequency injection methods presented in [17,18], the stator flux linkages of SMs with significant saliency are identified.The additionally injected currents require a specially designed current control system, lead to additional losses and may lead to undesired torque ripples (which is not acceptable for the commissioning of certain applications).Besides the approaches presented above, there exist several publications proposing observers for stator flux linkage estimation: In [19][20][21][22], sliding mode observers are proposed which require the induced voltage of the permanent magnets; and, hence are only suitable for (I/S)PMSMs.[23,24] propose a Gopinath style observer which is based on voltage and current dynamics but neglects cross-coupling.Only [25] takes cross-coupling effects into account by utilizing the integrated error of the current controller, but which implies again the need of a properly tuned current controller.Besides, disturbance observers are often used in machine control.In [26,27], the current is predicted with the help of an observer by considering the voltage error as disturbance.In [28], a disturbance observer for stator flux linkages and torque is proposed where the estimated disturbance is used for model predictive flux and torque control.
In contrast to the above, this paper proposes a novel disturbance observer that estimates the nonlinear stator flux linkages online.In addition to that, a simple sequence and postprocessing algorithm is proposed which allows to extract key information of the stator flux linkages within a pre-specified (d, q)-current range.The obtained data can be stored in lookup tables or used further for more sophisticated approaches like the fitting of prototype functions as proposed e.g. in [29].Hence, to the best knowledge of the authors, this work proposes the very first and simple disturbance observer based identification approach of nonlinear stator flux linkages which allows for (i) a (very) fast identification, (ii) application to any SM with constant or no excitation (e.g., S/IPMSM and RSM), and (iii) does not require a (current, voltage or speed) controller, a torque sensor or a prime mover.

II. STATOR FLUX LINKAGE DISTURBANCE OBSERVER A. Nonlinear synchronous machine model
The equivalent circuit of a nonlinear SM is shown in Fig. 1.The nonlinear flux linkage dynamics [7] are given by with stator voltages u dq s := (u d s , u q s ) ⊤ , stator currents i dq s := (i d s , i q s ) ⊤ , stator flux linkages ψ dq s = ψ dq s (i dq s , ω p ) and electrical speed ω p = n p ω m (i.e.pole pair number n p times mechanical angular velocity ω m ).Furthermore, the stator winding resistance matrix R dq s ∈ R 2×2 and rotation matrix J := 0 −1 1 0 .The stator flux linkages are nonlinear functions of the stator currents, i.e. ψ dq s = ψ dq s (i dq s ).Consequently, nonlinear cross-coupling and saturation effects are considered.

B. Observable state-space-model
The key idea for the disturbance observer design is to seperate the nonlinear flux linkages into a linear term L s,0 i dq s and a nonlinear flux linkage disturbance term ∆ dq ψ s (i dq s ), i.e.
The flux linkage dynamics are normalized to avoid illconditioned numerics.Therefore, the normalizations of system states and outputs, i.e.
where the following is defined to simplify notation Considering the nonlinear flux linkage disturbance term ∆ ψ as an additional disturbance state, the augmented system is with augmented state x, input u and output vector y as in For constant ω (as it is slowly varying compared to electrical quantities), the 4th order system (7) is observable if and only if the observability matrix which already shows that Q O (ω) has rank (Q O (ω)) ≥ 4 if and only if ω ̸ = 0 and rank (L) = 2. Consequently, the augmented system (3) is observable for all speeds ω = ω p ̸ = 0 and properly chosen (invertible) matrix L (which is a free design parameter, recall idea introduced in (2)).

C. Observer design
Assuming that stator resistance matrix R dq s and angular velocity ω = ω p are (exactly 1 ) known, leads to the observer of augmented system (7) with estimated state and output vector and feedback matrix F (ω) which must be designed to be speed-dependant because A(ω) as in (12) depends on ω as well.Furthermore, as system (7) represents a MIMO system which leads to more variables than equations in the observer design, the eigenstructure assignment described in [30] is beneficial to choose the feedback matrix F (ω) and allows to achieve pole placement of the four closed-loop observer poles.The relation between i-th left eigenvector v i and i-th eigenvalue λ i (equivalent to the i-th pole p i ) of the closed-loop observer matrix A(ω) − F (ω)C is given by which can be rewritten as where the replacement m i in ( 14) allows for a free eigenstructure assignment, e.g. the auxiliary vectors m i can be arbitrarily assigned (solely their linear independence must be guaranteed) with which the i-th eigenvector can be computed as follows For stability, the desired eigenvalues λ i (or poles) need to be chosen in the negative complex half-plane.Finally, collecting the computed eigenvectors v i (ω) and the auxiliary vectors m i of all four eigenvalues λ i , i ∈ {1, 2, 3, 4} in the matrices allows to compute the unique observer feedback matrix [30] As F (ω) in ( 17) establishes a Hurwitz observer matrix A(ω)− F (ω)C with the desired stable eigenvalues λ i , i ∈ {1, 2, 3, 4}, the estimation error e := x − x 1 A robustness analysis is crucial but considered future research here.
will decay exponentially to zero and the error dynamics are exponentially stable (a full proof for a similar system can be found in [31]).Please note that (18) also implies that the estimation errors e dq ψ s := ψ dq s − ψ dq s of flux linkages and e dq ∆ ψ := ∆ dq ψ s − ∆ dq ψ s of flux linkage disturbances tend to zero exponentially fast.

III. SIMULATIVE VALIDATION
The proposed flux linkage disturbance observer is implemented in Matlab & Simulink R2022a.For the simulation, a highly nonlinear, anisotropic IPMSM (taken from [7]) is utilized for the validation.Its key data is collected in Tab.I.
Ramp-like reference stator voltages are applied to the machine without the need of any current control system.The ramp-like voltages allow (i) to consider the transient behavior (of machine and observer) and (ii) to safely avoid overcurrents; since, if a pre-specified safety current threshold is reached, the voltages are simply reversed, stopped or set to zero.To improve the estimation performance of these ramplike signals, the observer model ( 12) must be extended by two integrator states (i.e., the internal model for ramp-like signals, see [32,Chapter 7.2]).The block diagram of the system and the extended observer with integral error feedback is shown in Fig. 2. The parameters of the implementation are listed in Table II.Since the observer feedback matrices F (ω) and F I (ω) depend on the angular velocity ω (recall e.g. ( 17)), only two exemplary gain matrices for ω = 100 rad s are given.The time series plots of the simulation results are shown in Fig. 3.The presented signals are, from top to bottom, the stator voltages u d s and u q s , the stator currents i d s and i q s , the estimated (observered) stator currents i d s and i q s , the current estimation errors e i d s = i d s − i d s and e i q s = i q s − i q s , the flux linkages ψ d s and ψ q s , the estimated flux linkages ψ d s and ψ q s , the flux linkage errors e ψ d s = ψ d s − ψ d s and e ψ q s = ψ q s − ψ q s , and the mechanical speed ω m = ω p n p .During the considered simulation scenario in Fig. 3, the nonlinear flux linkages can be estimated online and, after the experiment, estimates of the flux linkages can be extracted as data triples ( ψ d s , i d s , i q s ) and ( ψ q s , i d s , i q s ) and plotted into the flux linkage maps as illustrated in Fig. 4   The exemplary identification sequence in Fig. 3 has a duration of 1.2 s during which the stator voltages are ramped up and down until certain (pre-specified) current limits are reached.This sequence can be adjusted arbitrarily to the needs of the required operation range for which the flux linkages should/must be extracted.In this paper, one exemplary sequence is presented which allows to extract the flux linkage data along the d axis including the respective saturation region of the d flux linkage.A similar sequence could be applied to extract data along the q axis (omitted due to space limitations).
During the exemplary identification sequence in Fig. 3, the voltage u q s is ramped up during t = 0.1 s and t = 0.232 s and then ramped down again between t = 0.232 s and t = 0.364 s with an absolute rate of 500 V s .The sign of the ramp changes if the current limit |i d s | > 40 A or |i q s | > 40 A is reached.At the same time, the voltage u d s is ramped down during t = 0.1 s and t = 0.116 s with a slope of 500 V s until the maximum value u d s = −R s i max is reached, after which u d s is kept constant between t = 0.116 s and t = 0.347 s and, finally, is ramped down again to u d s = 0 V during t = 0.347 s and t = 0.364 s.These sequences for u d s and u q s are repeated during the remaining time interval until t = 1.2 s; during t = 0.364 s and t = 0.590 s with altered sign of u d s and during t = 0.590 s and t = 1.090 s with altered sign of u q s .As a consequence of the applied ramp-like voltages, the stator currents i d s and i q s and their estimates i d s and i q s are changing between −40 A and 40 A. The flux linkages ψ d s and ψ q s and their estimates ψ d s and ψ q s are varying between −50 mVs and 50 mVs.The estimation errors e i d s and e i q s of the stator currents and the estimation errors e ψ d s and e ψ q s of the flux linkages tend to zero very quickly (if the speed is not zero).
Due to the absence of any control system, the mechanical speed is changing during the identification sequence which can be exploited in a meaningful manner.As the machine is not observable at standstill (i.e., ω m = 0; recall (10)), the observer will only work properly for ω m ̸ = 0. To ensure a satisfactory estimation performance, a speed threshold of e.g.|ω m | > 40 rad s should be introduced.Then, for all speeds beyond this limit, the estimation error decays exponentially fast to zero.Moreover, to guarantee that the estimation errors e ψ d s and e ψ q s are small (e.g.smaller than e.g. 5% of the real values), a certain waiting time (here: 0.04 s) after exceeding the speed threshold should be considered before the data triples ( ψ d s , i d s , i q s ) and ( ψ q s , i d s , i q s ) can be stored with prescribed accuracy.The considered time intervals which were used for extimated flux linkage extraction are highlighted as gray areas in Fig. 3.
Finally, in Fig. 4, the real nonlinear flux linkage maps ψ d s (i d s , i q s ) and ψ q s (i d s , i q s ) of the considered machine are shown, whereas the estimated and extracted flux linkages ψ d s (i d s , i q s ) and ψ q s (i d s , i q s ), recorded during the identification sequence (recall gray area in Fig. 3

IV. CONCLUSION
A simple disturbance observer for online flux linkage estimation in combination with a simple identification sequence of ramp-like stator voltages has been proposed.It is applicable to any nonlinear synchronous machine (with constant or no excitation; e.g., RSMs, SPMSMs or IPMSMs).The disturbance observer allows to extract key information of the (d, q)flux linkages including saturation and cross-coupling effects as well as the permanent magnet flux linkage.Voltage ramps are applied to the machine and, thus, different operating points are reached within a second without the need for any controller (neither current, voltage nor speed controller) or additional hardware equipment (like torque sensor or prime mover).
Future research will (i) focus on a better and more intuitive observer tuning in order to obtain (even) smaller estimation errors and faster transients and (ii) elaborate on simpler identification sequences (e.g., using voltage steps instead of ramps).Moreover, the extracted flux linkage data will be used to estimate the whole flux linkage maps (e.g., by fitting physicsbased flux linkage prototype functions or neural networks).These estimates can then be used for current and/or optimal feedforward torque control.Finally, future experimental results will validate the performance and robustness of the proposed approach in the laboratory.

Figure 1 :
Figure 1: Equivalent circuit of the nonlinear SM model.

Figure 2 :
Figure 2: Extended observer with integral error feedback.

Figure 3 :
Figure 3: Times series plots of the observer implementation.
(a) d-axis flux linkage map with estimate.(a)q-axis flux linkage map with estimate.

Figure 4 :
Figure 4: Flux linkage maps of the nonlinear IPMSM with real stator flux linkages ψ d s(i d s , i q s ) & ψ q s (i d s , i q s ) [colored surface] and estimated stator flux linkages ψ d s (i d s , i q s ) & ψ q s (i d s , i q s ) [■].
Usually, the maximum values (e.g., maximum voltage u s,max , maximum current ı s,max and maximum speed ω m,max ) are (roughly) known and should be chosen for i norm = ı s,max and ψ norm = dq s := ψ norm ψ and i dq s := i norm i(4)with normalization factors i norm and ψ norm are introduced, respectively, in order to achieve that i and ψ are within the range of −1 to 1. d dt

Table II :
Exemplary observer parameters.